A ball rools off the top of a stairway with a horizontal velocity of magnitude 1.8 m/s . The steps are 0.20 m high and 0.2 m wide. Which step will the ball hit first?

To determine which step the ball will hit first, we need to analyze the motion of the ball as it rolls off the stairway. The ball has a horizontal velocity of 1.8 m/s and falls vertically due to gravity. We can break this problem into two parts: the horizontal motion and the vertical motion.

Understanding the Motion

The ball rolls off the stairway with a constant horizontal velocity. Since there are no horizontal forces acting on it (assuming air resistance is negligible), the horizontal motion can be described as:

• Horizontal Velocity (v_x): 1.8 m/s
• Horizontal Distance (d): This will depend on how long the ball is in the air.

For the vertical motion, the ball is influenced by gravity, which accelerates it downwards at approximately 9.81 m/s². The height of each step is 0.20 m, and we need to calculate how long it takes for the ball to fall this height.

Calculating the Time to Fall

To find out how long it takes for the ball to fall to the first step, we can use the following kinematic equation for vertical motion:

h = 0.5 * g * t²

Where:

• h: height (0.20 m for the first step)
• g: acceleration due to gravity (9.81 m/s²)
• t: time in seconds

Rearranging the equation to solve for time (t), we get:

t = √(2h/g)

Substituting the values:

t = √(2 * 0.20 m / 9.81 m/s²)

t ≈ √(0.0408) ≈ 0.202 seconds

Calculating Horizontal Distance

Now that we have the time it takes to fall to the first step, we can calculate how far the ball travels horizontally during this time:

d = v_x * t

Substituting the values:

d = 1.8 m/s * 0.202 s ≈ 0.364 m

Determining Which Step is Hit

Each step is 0.2 m wide, so we need to see how many steps the ball can travel horizontally before it hits the ground. The first step is at 0.2 m, the second step is at 0.4 m, and so on. Since the ball travels approximately 0.364 m horizontally, it will hit the second step, as it has traveled past the first step (0.2 m) but not quite to the third step (0.4 m).

Final Thoughts

In summary, the ball will hit the second step first after rolling off the stairway with a horizontal velocity of 1.8 m/s. This analysis combines both the vertical and horizontal motions to provide a clear understanding of the ball's trajectory. If you have any further questions or need clarification on any part of this explanation, feel free to ask!